W. Brown 
361 
(2) the close relation betweeu arithmetical ability and algebraical ability 
is due mainly to habits of accuracy common to both ; 
(3) memory of preceding propositions is the ability most closely related to 
the other forms of geometrical ability in school-boys, — is, in fact, the central ability 
in school geometry. 
It would be interesting to test the results for the existence of a general 
geometrical ability or a general mathematical ability, but unfortunately the only 
formula hitherto suggested for such a purpose, viz. that given by Dr Spearman*, 
is open to very serious criticism f. The writer hopes to return to the problem in 
a future paper. 
* Kruegerand Spearman : Zeitschrift filr PsycJwlogie, Bd. 44, 1906, p. 84. 
+ See Clark Wissler: "The Spearman Correlation Formula," Science, N. S. Vol. xxii. No. 558, 
Sept. 8, 1905, pp. 309—311. 
William Brown : "Some Experimental Eesults in Correlation," Comptes Rendus du Vl'"'^ Congres 
International de Psychologic, Geneve, Aug. 1909. In this pamphlet mention was made of the assumptions 
upon which the applicability of Spearman's formula was based, and an alternative proof of the formula 
based on those assumptions, contributed by Mr G. Udny Yule, was brought forward. 
The formula may be written 
where X and Y represent the unknown true values to be correlated, and A'l , A''2 ; Yi, Y'^ are two pairs of 
observed values, vitiated more or less by errors of observation. 
Let xi = x + 5i, yi = y + ei, 
X'2 = x + 8o, 2/3 = .'/ + e3, 
where x, y, 5, e represent deviations from means. 
The assumption involved in the formula is that the errors of measurement Sj, do, fi, e-, are un- 
correlated with each other or with x or y. 
I suggested that the justifiability of this assumption might be tested by correlating A'l ~ X-, and 
i'l ~ Y.,, Xi'-'X-i and Xi, Yi ~ Y^ and i'l which should all give a zero value for the coefficient. In this 
statement the sign of subtraction ( - ) should have been used instead of the difference-sign (~), and only 
the first of the three coefficients, viz. that between A'l - .Y2 and i'l - Y2, should be expected to give 
a zero value. 
Thus, r _ = Sj_{'-<'i-^2) {yi~yi)}_ 
n/s (5i-52FS(€i-e,)2 
= 0, since the numerator vanishes. 
Applying this test to some of the material on which I based my Geneva pamphlet, I found : — 
1. In the case of accuracy in bisecting and trisecting lines, where the subjects of the experiment 
were 43 adults (Group A in the paper) 
'b _^, = 0-30±0-09 ; 
t\- t\ 
2. In the ease of speed (S) and accuracy (A) in the addition of series of 10 single digits, where the 
subjects were 38 elementary school children, girls between the ages of 11 and 12 (Group C) 
'•^,-«„^.-A = 0-35±0-09. 
Thus, in each case a correlation greater than three times its probable error was found, showing that 
errors %vere almost certainly correlated among themselves and hence that Spearman's formula could not 
be applied. 
In cases, however, where the correlation does work out to zero we cannot infer that errors are 
not correlated, but only that 
S (5iei) + S (5262) = S (Sie.) + .S' (526,). 
The formula may still be quite inapplicable. 
